Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra

over a field is a Lie algebra

equipped with a nondegenerate skew-symmetric bilinear form

, which is a Lie algebra 2-cocycle of with values in . In other words,

:

for all , , in .

If is a coboundary, which means that there exists a linear form such that

then

is called a Frobenius Lie algebra.

Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form

If is a quasi-Frobenius Lie algebra, one can define on another bilinear product by the formula

:.

Then one has and

is a pre-Lie algebra.

See also

• Lie coalgebra
• Lie bialgebra
• Lie algebra cohomology
• Frobenius algebra
• Quasi-Frobenius ring

References

• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979.
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .